XIII
NonlinearandFractal
SignalProcessing
AlanV.Oppenheim
MassachusettsInstituteofTechnology
GregoryW.Wor nell
MassachusettsInstituteofTechnology
71ChaoticSignalsandSignalProcessing AlanV.OppenheimandKevinM.Cuomo
Introduction
•
ModelingandRepresentationofChaoticSignals
•
EstimationandDetection
•
Use
ofChaoticSignalsinCommunications
•
SynthesizingSelf-SynchronizingChaoticSystems
72NonlinearMaps StevenH.IsabelleandGregoryW.Wornell
Introduction
•
EventuallyExpandingMapsandMarkovMaps
•
SignalsFromEventuallyExpanding
Maps
•
EstimatingChaoticSignalsinNoise
•
ProbabilisticPropertiesofChaoticMaps
•
Statistics
ofMarkovMaps
•
PowerSpectraofMarkovMaps
•
ModelingEventuallyExpandingMapswith
MarkovMaps
73FractalSignals GregoryW.Wornell
Introduction
•
FractalRandomProcesses
•
DeterministicFractalSignals
•
FractalPointProcesses
74MorphologicalSignalandImageProcessing PetrosMaragos
Introduction
•
MorphologicalOperatorsforSetsandSignals
•
Median,Rank,andStackOperators
•
UniversalityofMorphologicalOperators
•
MorphologicalOperatorsandLatticeTheory
•
Slope
Transforms
•
MultiscaleMorphologicalImageAnalysis
•
DifferentialEquationsforContinuous-
ScaleMorphology
•
ApplicationstoImageProcessingandVision
•
Conclusions
75SignalProcessingandCommunicationwithSolitons AndrewC.Singer
Introduction
•
SolitonSystems:TheTodaLattice
•
NewElectricalAnalogsforSolitonSystems
•
CommunicationwithSolitonSignals
•
NoiseDynamicsinSolitonSystems
•
EstimationofSoliton
Signals
•
DetectionofSolitonSignals
76Higher-OrderSpectralAnalysis AthinaP.Petropulu
Introduction
•
DefinitionsandPropertiesofHOS
•
HOSComputationfromRealData
•
Linear
Processes
•
NonlinearProcesses
•
Applications/SoftwareAvailable
T
RADITIONALLY,SIGNALPROCESSINGasadisciplinehasreliedheavilyonatheoretical
foundationoflineartime-invariantsystemtheoryinthedevelopmentofalgorithmsfora
broadrangeofapplications.Inrecentyearsaconsiderablebroadeningofthistheoretical
basehasbeguntotakeplace.Inparticular,therehasbeensubstantialgrowthininterestintheuse
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1999byCRCPressLLC
of a variety of nonlinear systems with special properties for diverse applications. Promising new
techniquesforthe synthesis andanalysis of suchsystemscontinuetoemerge. Atthesametime, there
hasalso been rapidg rowth ininterestinsystemsthatarenot constrained tobetime-invariant. These
may be systems that exhibit temporal fluctuations in their characteristics, or, equally importantly,
systemscharacterizedbyotherinvarianceproperties,suchasinvariancetoscalechanges. Inthelatter
case, this gives rise to systems with fractal characteristics.
In some cases, these systems are directly applicable for implementing various kinds of signal
processingoperations such as signal restoration, enhancement, or encoding, or for modeling certain
kinds of distortion encountered in physical environments. In other cases, they serve as mechanisms
for generating new classes of signal models for existing and emerging applications. In particular,
whenautonomousordrivenbysimplerclassesofinputsignals,they generaterichclassesof signalsat
theiroutputs. Inturn,thesenewclassesofsignalsgiverisetonewfamiliesofalgorithmsforefficiently
exploiting them in the context of applications.
Thespectrum oftechniquesfornonlinear signalprocessingisextremelybroad,and in thischapter
we make no attempt to cover the entire array of exciting new directions b eing pursued within the
community. Rather, we present a ver y small sampling of several highly promising and interesting
ones to suggest the richness of the topic.
A br ief overview of the specific chapters comprising this section is as follows.
Chapters71and72discussthechaoticbehaviorofcertainnonlineardynamicalsystemsandsuggest
ways in which this behavior can be exploited. In particular, Chapter 71 focuses on continuous-time
chaoticsystems characterized bya special self-synchronizationproperty thatmakesthem potentially
attractive for a range of secure communications applications. Chapter 72 describes a family of
discrete-time nonlinear dynamical and chaotic systems that are particularly attractive for use in
a variety of signal processing applications ranging from signal modeling in power converters to
pseudorandom number generation and error-correctioncoding in signal transmission applications.
Chapter 73 discusses fractal signals which arise out of self-similar system models characterized by
scale-invariance. These represent increasingly important models for a range of natural and man-
made phenomena in applications involving both signal synthesis and analysis. Multidimensional
fractals also arise in thestate-spacerepresentationof chaotic signals, andthefractal propertiesin this
representationareimportantintheidentification,classification,andcharacterizationofsuchsignals.
Chapter 74 focuses on morphological signal processing, which encompasses an important class
of nonlinear filtering techniques together with some powerful associated signal representations.
Morphological signal processing is closely related to a number of classes of algorithms including
order-statisticsfiltering,cellularautomatamethodsforsignalprocessing,andothers. Morphological
algorithms are currently among the most successful and widely used nonlinear signal processing
techniques in image processing and vision for such tasks as noise suppression, feature extraction,
segmentation, and others.
Chapter 75 discusses the analysis and synthesis of soliton signals and their potential use in com-
munication applications. These signals arise in systems satisfying certain classes of nonlinear wave
equations. Because they propagate through those equations without dispersion, there has been
longstanding interest in their use as carrier waveforms over fiber-optic channels having the appro-
priate nonlinear characteristics. As they propagate through these systems, they also exhibit a special
type of reduced-energy superposition property that suggests an interesting multiplexing strategy for
communications over linear channels.
Finally, Chapter 76 discusses nonlinear representations for stochastic signals in terms of their
higher-order statistics. Such representations are particularly important in the processing of non-
Gaussian signals for which moretraditional second-moment characterizations are ofteninadequate.
The associated tools of higher-order spectral analysis find increasing application in many signal
detection, identification, modeling, and equalization contexts, where they have led to new classes of
powerful signal processing algor ithms.
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1999 by CRC Press LLC
Again,thesearticlesareonlyrepresentativeexamplesofthemanyemergingdirectionsinthisactive
areaofresearchwithinthesignalprocessingcommunity,anddevelopmentsinmanyotherimportant
and exciting directions can be found in the community’s journal and conference publications.
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1999 by CRC Press LLC
. richness of the topic.
A br ief overview of the specific chapters comprising this section is as follows.
Chapters71and72discussthechaoticbehaviorofcertainnonlineardynamicalsystemsandsuggest
ways. self-synchronizationproperty thatmakesthem potentially
attractive for a range of secure communications applications. Chapter 72 describes a family of
discrete-time